Fractional derivatives and periodic functions

Abstract

This paper studies the periodic functions in the perspective of fractional calculus application. It is shown that the fractional derivative of a periodic signal is periodic if it is defined on the whole real line. Several common fractional derivative formulations are considered, namely the Grunwald–Letnikov, Liouville and Caputo on $$\mathbb {R}$$ , and the two-sided fractional derivatives. It is verified that the fractional derivative of a causal periodic signal is never causal periodic. The periodic behaviour of the fractional linear systems is also studied. If such systems are defined with suitable derivatives the output corresponding to periodic input is also periodic. Is is concluded that only the integer order linear systems can have a sinusoidal impulse response.